Lesson video

Hello there, my name is Mr. Tilstone.

I hope you're having a good day today.

Let's see if we can make that day even better.

Today's lesson is all about coordinates.

This is probably not the first coordinates lesson you've done recently.

You are probably getting to be quite the coordinates expert.

Let's see if we can look at coordinates in term of reflection.

If you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is I can reflect simple shapes in the axes on a full coordinate grid.

So you might have had experience before with reflection, and you've hopefully got experience of coordinates.

Let's combine those skills together.

Our keywords today, my turn, axis, your turn.

And my turn, reflection, your turn.

Do you know what those words mean already? Shall we have a look? An axis is a reference line drawn on a graph.

The graph below has an x-axis and a y-axis.

The plural is axes, pronounced axes.

It looks like axes, but it's axes.

And a reflection is an image or shape as it would be seen in a mirror.

So every time you look in a mirror, you see your reflection.

And we've got an example here on a four quadrant coordinate grid.

Our lesson today is split into two cycles, two different parts.

The first will be reflect across one axis, and the second will be reflect across both axes.

Let's begin by thinking about reflecting across one axis.

In this lesson, you're going to meet Alex.

He's here today to give us a helping hand with the maths.

And here he is.

Alex is standing in front of a mirror.

What he sees is his reflection.

And maybe you've got to mirror in front of you right now.

Maybe you could use that and practise these skills along with Alex.

The closer he stands to the mirror, the closer his reflection is.

So let's see that.

Here we go, he stood a bit closer and his reflection appears closer.

And again.

And as I say, you might have a mirror in front of you, you could investigate that yourself now.

Get closer to the mirror and see what happens to your reflection.

Now, can you explain why these are not reflections of Alex? Let's take each of them in turn.

Can you explain why they're not reflections? None of these are his actual reflection.

I think some of them are a bit easier to explain than others, but have a go.

Let's start here.

This one's upside down.

That's not how your reflection would appear, is it, if you were standing in front of a mirror? Okay, what about the next one on the top right.

That doesn't look right, does it? What's the matter with that one? It's not the same size, so that's got to be wrong.

So what about this next one? Have a look at Alex's position next to the mirror and look at his reflection.

What do you notice there? It's not the same distance from the mirror, and it would have to be.

And what about the final one? What's wrong with that? Why is that not a reflection? He's facing the wrong way, so it can't be his reflection.

Now let's have a look at reflections on a four quadrant coordinate grid.

This transformation is an example of a reflection across the y-axis.

So you can see you've got a mirror there on the y-axis, and we're reflecting across the axis.

The shapes are congruent, and that's a word that hopefully you're getting more familiar with.

So the same shape, same size, and are at an equal distance from the y-axis.

Can you see that? So just like when you've got a mirror, your reflection is an equal distance from the mirror as you are, the same is true here on the four quadrant grid.

A mirror can be used to check.

You don't really need the mirror, but it's a good check.

See if you can explain whether or not the following transformations are reflections or not.

What about this one? Would you say that is a reflection? Think back to those examples with Alex and his mirror, and if it reminds you of any of those.

This transformation is not an example of a reflection across the y-axis.

Can you see why? The shapes are not at an equal distance from the y-axis, so they're not reflections.

So, no.

What about this one? What can we say here? Why is that not a reflection? 'Cause it's not.

It's not a reflection, even though the two shapes are congruent, but it's not a reflection.

Why? The shape has completed a rotation and translation, but not a reflection.

So, no, that's not a reflection.

What about this one? What can you notice here? Would you say that's a reflection or not? And if not, why not? No, it's not a reflection.

The two shapes are not congruent.

Can you see why? Look like they might be, but they're not quite.

The side lengths of one shape are longer than the other, so they are not congruent shapes, so that can't be a reflection.

So, no.

What about this one? Is that a reflection? If so, why? If not, why not? Nope, it's not a reflection.

The two shapes are congruent.

So it's got one thing going for it, but it's not a reflection because what? It's a translation, but it's not a reflection.

It's been translated across but not reflected.

So, nope.

One of the shapes needs to be flipped to make it a reflection.

A reflection appears like a translation that's been flipped.

So if you can imagine translating something and then flipping it, you've got a reflection.

So this can be shown with tracing paper.

So let's put some tracing paper on there.

And you might have some tracing paper in your classroom.

You could do this when you do your reflections later on.

So we've traced over that triangle, and of course what we've traced is congruent.

And now we can move it over, we can translate it, but it's not reflected yet.

It's been translated but not reflected.

Now we flip it over and yes, now it is a reflection.

Plotting each reflected vertex and joining them together is an efficient way of forming a reflection.

This vertex is two units from the y-axis.

Its reflection must also be two units from the y-axis.

Let's have a look at that.

Here we go.

So they're the same distance from the axis.

The other two vertices are three units from the y-axis.

So their reflections must be three units from the y-axis, and we can plot those in by counting.

That's away away and that's three away.

Now all we've gotta do is join the vertices together to make our reflection.

And now we've got a reflection.

The same shape is now being reflected across the x-axis.

So these vertices are each one unit from the x-axis.

Can you picture, can you visualise, where they will be on this four quadrant grid if they're being reflected across the x-axis? So imagine that mirror now on the x-axis, not the Y.

Where are the reflected vertices going to be? There, just there.

And what about this one? This is now three units from the x-axis, and so will its reflection be.

So counting from the x-axis, counting three away and you land just here, and we can join those together.

And we've got a reflection this time in the x-axis.

Let's do a quick check for understanding.

Has this shape been reflected across the x-axis, and can you explain? Pause the video and have a go.

Let's have a look at this.

Did you manage to come up with an agreement with the people around you? Has this been reflected across the x-axis? No, it hasn't.

It has been translated, but it hasn't been reflected.

The vertices are different distances from the mirror line.

The shape needs to be flipped so we can sort it out.

Here we go, and now if you look at any vertex, it's the same distance from the x-axis as its reflection.

Let's have a look at another example.

When reflected across the x-axis, what do you notice? Let's think about this.

So we've got -3, 1 one for the original.

Can you see that, which vertex that would be? And the reflection is -3, -1.

And in the original -3, 3 is a coordinate, a vertex and its reflection is -3, -3.

And then finally we've got a vertex at -2, 1, and the reflection is -2, -1.

Now what did you notice there? Did you notice anything? The X coordinate does not change.

The Y coordinates are the same, but one is positive and one is negative.

Did you spot that? So the 1 became -1, the 3 became -3, and the 1 became -1.

So when reflected across the y-axis, what do you notice? Let's have a look at the original.

<v ->3, 1,</v> 3, 1.

<v ->3, 3,</v> 3, 3.

<v ->2, 1,</v> 2, 1.

Did you notice some things are the same there and some things were different? The Y coordinate, in this case, does not change.

The X coordinates are the same, but one is positive and one is negative.

So in the original it was -3, but in the reflection 3, for example.

Let's have a check, see if you've understood that part.

A shape has been reflected across the x-axis.

What are the coordinates of the reflection? So they are the original coordinates.

the original vertices, -4, 2, -2, and -4, 1.

What will be the reflection? Can you picture that? Can you visualise that? Can you use your skills? Can you remember? Have a go.

Pause the video.

Let's see.

Well, the X coordinates remain unchanged, but the Y coordinates have changed from positive to negative.

So in this case we've got -4 as it was before, but -2.

So 2 has gone from positive 2 to -2.

And in the original -2, 2 becomes -2, -2.

And in the original -4, 1, the reflection is -4, -1.

So very well done if you managed to get those.

It's time for some practise.

Number one, reflect the shapes across the x-axis.

So the x-axis, make sure you're using the right axis here.

And if you look at C, there's something a bit unusual about C.

I won't say what it is.

Something you might need to do a little bit of thinking about.

Have you spotted it? And number two, reflect the shapes across the y-axis.

And some of them are more complicated than others, particularly the ones that have got diagonal lines in, as these have.

But my top tip is think about how far each coordinate is from the axis.

And the same for C.

Have a look at C again, think how that might change.

And number three, a shape has been reflected across the x-axis.

What is the shape and what are the coordinates of the reflection? So there are the original coordinates.

What would be the coordinates of the reflection across the x-axis and what's the shape? Number four, a different shape has been reflected across the y-axis.

What is the shape and what are the coordinates of the reflection? Okay, very best of luck with that.

If you can work with somebody else, I always recommend that.

Then you can share ideas with each other.

Pause the video and I will see you soon for some feedback.

Welcome back, how did you get on with Cycle A? Would you like some answers? You've got it.

Here we go.

So number one, reflect the shapes across the x-axis.

That's a reflection of shape A, and this is a reflection of shape B.

So well done if you plotted those in the right quadrant and well done if you were accurate.

And this is C, so this is a reflection of C.

And number two, reflect the shapes across the y-axis this time.

This is a reflection of shape A, and this is a reflection of shape B.

And this is a slightly tricky one, I think, because the zero Y coordinates are still zero Y coordinates after the reflection as they lie directly on the axis.

So if you can imagine your nose being pressed to a mirror, your reflection would appear the same way, wouldn't it? So I think it's similar to that.

So that's the reflection.

Number three, a shape's been reflected across the x-axis.

What is the shape and what are the coordinates of the reflection? Well, that's the shape.

It's a rectangle.

And in terms of the reflection of its coordinates, the original 1, 1 becomes 1, -1.

So the X coordinate did not change, but the Y coordinate went from positive to negative.

And that's the case with all of these.

So then we've got 1, -5, 4, -1, and 4, -5.

And that's what the reflection would look like.

When reflecting across the x-axis, The X values remain unchanged, but the Y values change from positive to negative or negative to positive depending on which quadrant they start in.

And the same question again with number four, different shape.

So this is the shape and these are the coordinates of the reflection.

It's a trapezium and we go -2, -1.

So in this case, the Y coordinate is unchanged, but the X coordinate goes from positive to negative, in this case.

<v ->4, -1,</v> <v ->1, -4,</v> and -5, -4.

And that's where the reflection would be.

When reflecting across the y-axis, the Y values remain unchanged, but the X values change from positive to negative or negative to positive, depending on where they start.

Hey, you are doing great here.

Let's move on to cycle B.

Let's reflect across both axes.

"If you reflect a shape", says Alex, "In an axis, and then reflect it again, it appears back in its original position.

It looks like there is no reflection." Do you agree with Alex? Hmm? So, so far we've just been reflecting across one axis.

Let's see if we agree with him or not.

What do you think? Well, that would be true if the shape were reflected back across the same axis.

The reflection of the reflection would be the original shape.

So sometimes that's true.

Let's see that in action.

So we've got a shape here, so we're going to reflect it across the y-axis, and then, that's a reflection of it.

Then we're going to reflect it back across the y-axis.

Yes, and it appears back in it's original position.

However, can you visualise where the reflection would be if this shape were reflected across both axes? And that's something we've not done so far.

Let's have a look.

Well, it does not matter which axis you choose first.

So you could reflect it across the x-axis and then the y-axis, or you could reflect it across the y-axis and then the x-axis, it does not matter.

This is across the y-axis.

So that's the first reflection, and now we're going to reflect that reflection, but this time across the x-axis, where do you think it would appear? This reflection then reflected across the other axis to complete that transformation.

And this is how the final reflection would appear.

So that shape, the original shape, has been reflected across two axes.

When reflecting across the other axis first, the final reflection will still be in the same place.

So it does not matter which way you start.

What do you notice about the coordinates of the reflection? Let's think about that.

Let's look at the original, think about the original coordinates and let's think about the coordinates of the reflection.

Well, that's the original, -3, 1 became 3, -1.

Hmm.

Did you notice something there? <v ->3, 3 became 3, -3.

</v> I'm definitely seeing a pattern here.

And -2, 1 became 2, -1.

What could you say about those coordinates? You could say each negative value has become positive and each positive value has become negative.

So -3 became 3, <v ->3 became 3,</v> and -2 became 2, 1 became -1, 3 became -3 and 1 became -1.

Let's have a little check.

Has the shape been reflected across both axes? Can you explain that? Pause the video.

Well, there was lots of ways you could approach this.

You could think about the coordinates, for example.

But the answer is no, the shape is facing the wrong way.

Did you spot that? And it's too close to the y-axis.

So there was two things not right about that reflection.

It's not really a reflection at all.

So let's look at those coordinates.

So this is the original 1, 1, and then -2, -1 for this so-called reflection, it's not really a reflection.

Yeah, 3, 1, -2, -3, 3, 3, 0, -3.

1, 3 became 0, -1 and 2, 2 became -1, -2.

Well, that doesn't really match up with what we just explored with the other example, does it, where the positive coordinates became negative and the negative became positive? So something's not right here.

If this was a reflective shape, each negative value would be positive and each positive value would be negative.

So we can use that to help us plot out the correct reflections.

Let's see.

So the original 1, 1 should be -1, -1.

Both those positive values should become negative.

The original 3, 1 should be -3, -1.

The original 3, 3, -3 -3.

The original 1, 3 should be -1, -3.

The original 2, 2, should be -2, -2.

So all of the coordinates in that first example were positive, so therefore, all of the coordinates in the reflected example reflected across both axis, should be negative.

And that is the correct position.

That looks right, doesn't it? That's the right distance from the axis and it's in the right position, the right orientation.

So yes, that's correct.

It is time for some practise.

Is Alex correct? He says, "When reflecting your shape across both axes always reflect in the x-axis first." Well, do you agree? Is that right? Can you explain? Number two, reflect the shapes across both axes this time, both axes.

And you've got a few examples there.

Again, there might be something you notice about D that makes it a little bit trickier.

Number three, a shape has been reflected across both axes.

What's the shape and what are the coordinates of the reflection? So I can see three coordinates.

That's a little clue to start with.

And then remember, it's been reflected across both axes.

So think about what you're going to need to do to those coordinates.

So good luck with all of that.

Pause the video and off you go.

Welcome back, how did you get on with that? Let's have a look.

This was reflecting across both axis.

So Alex was incorrect.

It doesn't matter whether you first reflect in the x-axis and then reflect the reflection in the y-axis or vice versa.

It doesn't matter.

Either way, the reflection will be in the quadrant diagonally opposite the starting point.

So it really doesn't matter.

He wasn't right there.

And then here are the reflections across both axis.

And it doesn't matter which axis you reflect across first, both possibilities are shown before the final reflection is given.

So it's where the final reflection is that really matters.

Here we go.

So that's the reflection across the y-axis.

And then that's the reflection of the reflection across the x-axis.

So that's where that should be.

And like I say, we could do that the other way around.

Maybe you did.

We could reflect across the x-axis first, like so, and then the y-axis second.

But we still end up with the reflection of the reflection in the exact same position.

And then for B, you could go this way first, across the y-axis, and then reflect that.

Or you could reflect across the x-axis and then reflect the reflection across the y-axis.

But either way, you end up with the reflection of the reflection just here.

So well done if you've got that.

And for C, bit of a tricky shape C was, wasn't it? It had lots of vertices and a diagonal line in there as well.

It was a bit of a tricky one.

Let's see how you got on.

So that is where the reflection of the reflection would be.

And you might have done it that way round, but it's still in the same position.

And then for D, it was on the x-axis, one of the sides was, two of the vertices.

So that's a reflection across the x-axis.

And then that's a reflection across the y-axis.

So very well done if you got that.

I think that was the trickiest one so far.

You could have done it this way round as well.

Doesn't matter.

And then a shape's been reflected across both axes.

What's the shape? What are the coordinates of the reflection? That's the shape.

It's a scalene triangle.

And the reflection, -1, -1, <v ->2, -4,</v> and -5, -1.

So all three of those positive values became negative values.

And this is what it would look like reflected on the four quadrant grid.

So here's a reflection across the x-axis, and here's a reflection of the reflection across the y-axis just here, just like that.

We've come to the end of the lesson.

I've really enjoyed exploring this concept with you today.

Today's lesson has been reflecting simple shapes in the axes on a full coordinate grid.

When reflecting shapes in a four quadrant coordinate grid, the axes can be thought of as mirror lines and the shape can be reflected in one or both of them.

And you've done both.

In cycle A, it was across one of the axes, and in cycle B, it was across both axes.

Counting from each vertex to the mirror line and then back from the mirror line to create the reflected vertex is one strategy.

And I like that one.

That's my favourite strategy.

Considering the coordinates of the original and reflective shapes is a different way to do it.

And then you can plot in those as soon as you know them, as soon as you've worked those out.

Very well done on your achievements today.

You've been absolutely amazing.

I hope I get the chance to spend another math lesson with you in the near future.

But until then, enjoy the rest of your day, whatever you've got in store.

Take care and goodbye.